Find the Maximum/Minimum Value f(x)=3(x-1)^2+2

$f (x) = 3 {(x - 1)}^{2} + 2$

Step 1

Simplify each term.

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$f (x) = 3 ((x - 1) (x - 1)) + 2$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$f (x) = 3 (x (x - 1) - 1 (x - 1)) + 2$

Apply the distributive property.

$f (x) = 3 (x \cdot x + x \cdot - 1 - 1 (x - 1)) + 2$

Apply the distributive property.

$f (x) = 3 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) + 2$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = 3 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) + 2$

Move $- 1$ to the left of $x$ .

$f (x) = 3 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) + 2$

Rewrite $- 1 x$ as $- x$ .

$f (x) = 3 (x^{2} - x - 1 x - 1 \cdot - 1) + 2$

Rewrite $- 1 x$ as $- x$ .

$f (x) = 3 (x^{2} - x - x - 1 \cdot - 1) + 2$

Multiply $- 1$ by $- 1$ .

$f (x) = 3 (x^{2} - x - x + 1) + 2$

Subtract $x$ from $- x$ .

$f (x) = 3 (x^{2} - 2 x + 1) + 2$

Apply the distributive property.

$f (x) = 3 x^{2} + 3 (- 2 x) + 3 \cdot 1 + 2$

Simplify.

Multiply $- 2$ by $3$ .

$f (x) = 3 x^{2} - 6 x + 3 \cdot 1 + 2$

Multiply $3$ by $1$ .

$f (x) = 3 x^{2} - 6 x + 3 + 2$

Step 2

Add $3$ and $2$ .

$f (x) = 3 x^{2} - 6 x + 5$

Step 3

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 4

Find the value of $x = - \frac{b}{2 a}$ .

Substitute in the values of $a$ and $b$ .

$x = - \frac{- 6}{2 (3)}$

Remove parentheses.

$x = - \frac{- 6}{2 (3)}$

Simplify $- \frac{- 6}{2 (3)}$ .

Cancel the common factor of $- 6$ and $2$ .

Factor $2$ out of $- 6$ .

$x = - \frac{2 \cdot - 3}{2 \cdot 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 3$ .

$x = - \frac{2 \cdot - 3}{2 (3)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 3}{2 \cdot 3}$

Rewrite the expression.

$x = - \frac{- 3}{3}$

Cancel the common factor of $- 3$ and $3$ .

Factor $3$ out of $- 3$ .

$x = - \frac{3 \cdot - 1}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$x = - \frac{3 \cdot - 1}{3 (1)}$

Cancel the common factor.

$x = - \frac{3 \cdot - 1}{3 \cdot 1}$

Rewrite the expression.

$x = - \frac{- 1}{1}$

Divide $- 1$ by $1$ .

$x = - - 1$

Multiply $- 1$ by $- 1$ .

$x = 1$

Step 5

Evaluate $f (1)$ .

Replace the variable $x$ with $1$ in the expression.

$f (1) = 3 {(1)}^{2} - 6 \cdot 1 + 5$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 3 \cdot 1 - 6 \cdot 1 + 5$

Multiply $3$ by $1$ .

$f (1) = 3 - 6 \cdot 1 + 5$

Multiply $- 6$ by $1$ .

$f (1) = 3 - 6 + 5$

Simplify by adding and subtracting.

Subtract $6$ from $3$ .

$f (1) = - 3 + 5$

Add $- 3$ and $5$ .

$f (1) = 2$

The final answer is $2$ .

$2$

Step 6

Use the $x$ and $y$ values to find where the minimum occurs.

$(1, 2)$

Step 7

Find the Maximum/Minimum Value f(x)=3(x-1)^2+2

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